Discrete and Continuous Dynamical Systems - B
August 2002 , Volume 2 , Issue 3
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Evolution equations that feature both nonlinear and dispersive effects often possess solitary-wave solutions. Exact theory for such waves has been developed and applied to single equations of Korteweg-de Vries type, Schrödinger-type and regularized long-wave-type for example. Much less common has been the analysis of solitary-wave solutions for systems of equations. The present paper is concerned with solitary travelling-wave solutions to systems of equations arising in fluid mechanics and other areas of science and engineering. The aim is to show that appropriate modification of the methods coming to the fore for single equations may be effectively applied to systems as well. This contention is demonstrated explicitly for the Gear- Grimshaw system modeling the interaction of internal waves and for the Boussinesq systems that arise in describing the two-way propagation of long-crested surface water waves.
A uniform lower bound for the energy involved in the propagation of a flame is given. Such a bound is important for safety considerations. In the integro-differential model the spherical flame originates from a point source which supplies a finite amount of energy over time. It is proved here that, independently of the form of the heat source function, a minimal energy is required for the propagation of the flame. The effect of a spark is then studied.
We prove existence of quasiperiodic breathers in Hamiltonian lattices of weakly coupled oscillators having some integrals of motion independent of the Hamiltonian. The proof is obtained by constructing quasiperiodic breathers in the anticontinuoum limit and using a recent theorem by N.N. Nekhoroshev  as extended in  to continue them to the coupled case. Applications to several models are given.
We study a traffic flow model with inhomogeneous road conditions such as obstacles. The model is a system of nonlinear hyperbolic equations with both relaxation and sources. The flux and the source terms depend on the space variable. Waves for such a system propagate in a more complicated way than those do for models with homogeneous road conditions.
The $L^1$ well-posedness theory for the model is established. In particular, we derive the continuous dependence of the solution on its initial data in $L^1$ topology. Moreover, the $L^1$-convergence to the unique zero relaxation limit is proved. Finally, the asymptotic states of a general solution whose initial data tend to constant states as $|x| \rightarrow +\infty$ are constructed.
We study the logistic map $f(x)=\lambda x (1-x)$ on the unit square at the chaos threshold. By using the methods of symbolic dynamics, the information content of an orbit of a dynamical system is defined as the Algorithmic Information Content (AIC) of a symbolic sequence. We give results for the behaviour of the AIC for the logistic map. Since the AIC is not a computable function we use, as approximation of the AIC, a notion of information content given by the length of the string after it has been compressed by a compression algorithm, and in particular we introduce a new compression algorithm called CASToRe. The information content is then used to characterise the chaotic behaviour.
We investigate the reduction of complex chemistry in gaseous mixtures. We consider an arbitrarily complex network of reversible reactions. We assume that their rates of progress are given by the law of mass action and that their equilibrium constants are compatible with thermodynamics; it thus provides an entropic structure  . We study a homogeneous reactor at constant density and internal energy where the temperature can encounter strong variations. The entropic structure brings in a global convex Lyapounov function and the well-posedness of the associated finite dimensional dynamical system. We then assume that a subset of the reactions is constituted of "Fast" reactions. The partial equilibrium constraint is linear in the entropic variable and thus identifies the "Slow" and "Fast" variables uniquely in the concentration space through constant orthogonal projections. It is proved that there exists a convex compact polyhedron invariant by the dynamical system which contains an affine foliation associated with a Tikhonov normal form. The reduction step is then identified using the orthogonal projection onto the partial equilibrium manifold and proved to be compatible with the entropy production. We prove the global existence of a smooth solution and of an asymptotically stable equilibrium state for both the reduced system and the complete one. A global in time singular perturbation analysis proves that the reduced system on the partial equilibrium manifold approximates the full chemistry system. Asymptotic expansions are obtained.
We consider the break-up of invariant tori in Hamiltonian systems with two degrees of freedom with a frequency which belongs to a cubic field. We define and construct renormalization-group transformations in order to determine the threshold of the break-up of these tori. A first transformation is defined from the continued fraction expansion of the frequency, and a second one is defined with a fixed frequency vector in a space of Hamiltonians with three degrees of freedom.
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